Minor things (where they don't start shooting you immediately) you don't have to run away. I had one occasion where I got busted robbing someone. I surrendered to the lawman that caught me and he said it wasn't worth his time but I should leave. I walked away slowly at first and he shouted something along the lines of "GTFO". I was curious what he would've done had I stayed, but I didn't want to bother at the time.
You're unable to surrender when they're shooting, they just want you dead at that point.
I've had 3 different scenes play out after being arrested. Typically you'll just serve your time and be released. Sometimes the gang will come and break you out (it goes much better than other jailbreaks you might have encountered). The third is (I'm assuming) more rare. I've only seen it once and never heard anyone else mention it. I bribed the lawman that was watching me and he let me go. Not sure if it actually cost me anything. You don't have a choice (the scenes play out randomly), so I don't think they actually take your money.
Not sure about being wanted dead or alive, but if they don't immediately open fire then you should have the option to surrender.